Contemporary Mathematics

Two exact sequences for lattice cohomology

András Némethi

Dedicated to Henri Moscovici on his 65th birthday.

Abstract. This article is a continuation of [N08], where the lattice cohomology of connected negative definite plumbing graphs was introduced. Here we consider the more general situation of non–degenerate plumbing graphs, and we establish two exact sequences for their lattice cohomology. The first is the analogue of the surgery exact triangle proved by Ozsváth and Szabó for the Heegaard–Floer invariant HF^{+}; for the lattice cohomology over ℤ_{2}– coefficients it was proved by J. Greene in [Gr08]. Here we prove it over ℤ, and we supplement it by some additional properties valid for negative definite graphs. The second exact sequence is an adapted version which does not mix the classes of the characteristic elements (spin^{c}–structures); it was partially motivated by the surgery formula for the Seiberg–Witten invariant obtained in [BN10]. For this, we define the relative lattice cohomology and we determine its Euler characteristic in terms of Seiberg–Witten invariants.

1. Introduction

The lattice cohomology {ℍ^{q}(Γ)}_{q≥0} was introduced in [N08]. In its original version, it was associated with any connected negative definite plumbing graph Γ, or, equivalently, with any oriented 3–manifold, which might appear as the link of a local complex normal surface singularity. Lattice cohomology (together with the graded roots) plays a crucial role in the comparison of the analytic and topological invariants of surface singularities, cf. [N05, N07, N08], see also [BN10, NN02, N10] for relations with the Seiberg–Witten invariants of the link. Additionally, the lattice cohomology (conjecturally) offers a combinatorial description for the Heegaard–Floer homology of Ozsváth and Szabó (for this theory see [OSz03, OSz04, OSz04b] and the long list of articles of Ozsváth and Szabó). Indeed, in [N08] the author conjectured that ⊕_{q even/odd}ℍ^{q}(Γ) is isomorphic as

2010 Mathematics Subject Classification. Primary. 32S25, 57M27. Secondary. 14B05, 14J17, 32S45, 57R57.

Key words and phrases. plumbed manifolds, plumbing graphs, 3-manifolds, lattice cohomology, surface singularities, rational singularities, Seiberg-Witten invariant, Heegaard-Floer homology.

The author is partially supported by OTKA Grant K67928.

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

a graded ℤ[U]–module with HF_{even/odd}^{+}(−M(Γ)), where M(Γ) is the plumbed 3– manifold associated with Γ. (Recall that at this moment there is no combinatorial definition/characterization of HF^{+}.)

For rational and ‘almost rational’ graphs this correspondence was established in [OSz03, N05] (see also [NR10] for a different situation and [N08] for related results). A possible machinery which might help to prove the general conjecture is based on the surgery exact sequences. They are established for the Heegaard–Floer theory in the work of Ozsváth and Szabó. Our goal is to prove the analogous exact sequences for the lattice cohomology. In fact, independently of the above conjecture and correspondence, the proof of such exact sequences is of major importance, and they are fundamental in the computation and in finding the main properties of the lattice cohomologies.

The formal, combinatorial definition of the lattice cohomology permits one to extend its definition to arbitrary graphs (plumbed 3–manifolds); the connectedness and negative definiteness assumptions can be dropped. Nevertheless, in the proof of the exact sequences, we will deal only with non–degenerate graphs (they are those graphs whose associated intersection form has non–zero determinant).

More precisely, for any graph Γ and fixed vertex j_{0}, we consider the graphs Γ\j_{0} and Γ_{j_{0}}^{+}, where the first one is obtained from Γ by deleting the vertex j_{0} and adjacent edges, while the second one is obtained from Γ by replacing the decoration e_{j_{0}} of the vertex j_{0} by e_{j_{0}} + 1. We will assume that all these graphs are non–degenerate. Then Theorem 5.5.3 establishes the following long exact sequence.

Theorem A. Assume that the graphs Γ_{j_{0}}^{+}, Γ and Γ \ j_{0} are non–degenerate. Then

⋯ ⟶ ℍ^{q}(Γ_{j_{0}}^{+}) ⟶^{𝔸^{q}} ℍ^{q}(Γ) ⟶^{\overline{𝔹}^{q}} ℍ^{q}(Γ \ j_{0}) ⟶^{ℂ^{q}} ℍ^{q+1}(Γ_{j_{0}}^{+}) ⟶ ⋯

is an exact sequence of ℤ[U]–modules.

The first 3 terms of the exact sequence (i.e. the ℍ^{0}–part) were already used in [OSz03] (see also [N05]), and the existence of the long exact sequence was already proved over ℤ_{2}–coefficients in [Gr08]. Here we establish its validity over ℤ. In the proof we not only find the correct signmodifications, but we also replace some key arguments. (Nevertheless, the proof follows the main steps of [Gr08].)

For negative definite graphs (i.e. when Γ_{j_{0}}^{+}, hence Γ and Γ \ j_{0} too are negative definite), the above exact sequence has some important additional properties. By general theory (cf. [N08]), if Γ is negative definite then ℍ^{0}(Γ) contains a canonical submodule 𝕋 and one has a direct sum decomposition ℍ^{0} = 𝕋 ⊕ ℍ_{red}^{0}. 𝕋 is the analogue of the image of HF^{∞} in HF^{+} in the Heegaard–Floer theory. On the other hand, in Heegaard–Floer theory, by a result of Ozsváth and Szabó, the operator ℂ^{0} restricted to 𝕋(Γ \ j_{0}) is zero (this follows from the fact that the corresponding cobordism connecting Γ \ j_{0} and Γ_{j_{0}}^{+} comes from a non–negative definite surgery). The analogue of this result is Theorem 6.1.2:

Theorem B. Assume that Γ_{j_{0}}^{+} is negative definite. Consider the exact sequence

0 ⟶ ℍ^{0}(Γ_{j_{0}}^{+}) ⟶^{𝔸^{0}} ℍ^{0}(Γ) ⟶^{\overline{𝔹}^{0}} ℍ^{0}(Γ \ j_{0}) ⟶^{ℂ^{0}} ℍ^{1}(Γ_{j_{0}}^{+}) ⟶ ⋯

and the canonical submodule 𝕋(Γ\j_{0}) of ℍ^{0}(Γ\j_{0}). Then the restriction ℂ^{0}|𝕋(Γ\j_{0}) is zero.

Using the above results one proves for a graph Γ with at most n bad vertices (for the definition, see 6.2) the vanishing ℍ_{red}^{q}(Γ) = 0 for any q ≥ n (where ℍ_{red}^{q} = ℍ^{q} for q > 0).

ℍ^{∗}(Γ) has a natural direct sum decomposition indexed by the set of the characteristic element classes (in the case of the Heegaard–Floer, or Seiberg–Witten theory, they correspond to the spin^{c}–structures of M(Γ)). Namely, ℍ^{∗}(Γ) = ⊕_{[k]}ℍ^{∗}(Γ, [k]). In the exact sequence of Theorem A the operators mix these classes. Theorem 7.2.2 provides an exact sequence which connects the lattice cohomologies of Γ and Γ\j_{0} with fixed (un–mixed) characteristic element classes. More precisely, for any characteristic element k of Γ, we define a ℤ[U]–module {ℍ_{rel}^{q}(k)}_{q≥0}, the relative lattice cohomology associated with (Γ, j_{0}, k). It has finite ℤ–rank and it fits in the following exact sequence:

Theorem C. Assume that Γ and Γ \ j_{0} are non–degenerate. One has a long exact sequence of ℤ[U]–modules:

⋯ ⟶ ℍ_{rel}^{q}(k) ⟶^{𝔸_{rel}^{q}} ℍ^{q}(Γ, [k]) ⟶^{𝔹_{rel}^{q}} ℍ^{q}(Γ \ j_{0}, [R(k)]) ⟶^{ℂ_{rel}^{q}} ℍ_{rel}^{q+1}(k) ⟶ ⋯

where R(k) is the restriction of k (and the operators also depend on the choice of the representative k).

The existence of such a long exact sequence is predicted and motivated by the surgery formula for the Seiberg–Witten invariant established in [BN10]: the results of section 7 resonate perfectly with the corresponding statements of Seiberg–Witten theory. This allows us to compute the Euler characteristic of the relative lattice cohomology in terms of the Seiberg–Witten invariants associated with M(Γ) and M(Γ \ j_{0}).

2. Notations and preliminaries

2.1. Notations. First we will introduce the needed notations regarding plumbing graphs and we recall the definition of the lattice cohomology from [N08]. Since in [loc.cit.] the graphs were connected and negative definite (as the ‘normal’ plumbing representations of isolated complex surface singularity links), at the beginning we will start with these assumptions.

Let Γ be such a plumbing graph with vertices J and edges E; we set |J | = s and we fix an order on J. Γ can also be codified in the lattice L, the free ℤ–module generated by {E_{j}}_{j∈J} and the ‘intersection form’ {(E_{i}, E_{j})}_{i,j}, where (E_{i}, E_{j}) for i ≠ j is 1 or 0 according as whether (i, j) is an edge or not; and (E_{i}, E_{i}) is the decoration of the vertex i, usually denoted by e_{i}. (The graph is negative definite if this form is so.) The graph Γ may have cycles, but we will assume that all the genus decorations are zero (i.e. we plumb S^{1}–bundles over S^{2}). The associated plumbed 3–manifold M(Γ) is not necessarily a rational homology sphere; this happens exactly when the graph is a tree. Let L′ be the dual lattice {l′ ∈ L ⊗ ℚ : (l′, L) ⊂ ℤ}; it is generated by {E_{j}^{∗}}_{j}, where (E_{j}^{∗}, E_{i}) = −δ_{ij} (the negative of the Kronecker delta). Moreover,

Char := \{k ∈ L′ : χ_{k}(l) := −\frac{1}{2}(k + l, l) ∈ ℤ for all l ∈ L\}

denotes the set of characteristic elements of L (or Γ).

Following Ozsváth and Szabó, we set T_{0}^{+} for the ℤ[U]–module ℤ[U, U^{−1}]/Uℤ[U] with grading deg(U^{−d}) = 2d (d ≥ 0). More generally, for any r ∈ ℚ one defines T_{r}^{+}, the same module as T_{0}^{+}, but graded (by ℚ) in such a way that the (d + r)– homogeneous elements of T_{r}^{+} are isomorphic to the d–homogeneous elements of T_{0}^{+}. (E.g., for m ∈ ℤ, T_{2m}^{+} = ℤ[U, U^{−1}]/U^{−m+1}ℤ[U].)

2.2. The lattice cohomology associated with k ∈ Char [N08]. L ⊗ ℝ = ℤ^{s}⊗_{ℤ}ℝ has a natural cellular decomposition into cubes. The set of zero–dimensional cubes is provided by the lattice points L. Any l ∈ L and subset I ⊂ J of cardinality q defines a q–dimensional cube, which has its vertices in the lattice points (l + ∑_{j∈I′} E_{j})_{I′}, where I′ runs over all subsets of I. On each such cube we fix an orientation. This can be determined, e.g., by the order (E_{j_{1}}, …, E_{j_{q}}), where j_{1} < ⋯ < j_{q}, of the involved base elements {E_{j}}_{j∈I}. The set of oriented q–dimensional cubes defined in this way is denoted by Q_{q} (0 ≤ q ≤ s).

Let C_{q} be the free ℤ–module generated by oriented cubes □_{q} ∈ Q_{q}. Clearly, for each □_{q} ∈ Q_{q}, the oriented boundary ∂□_{q} has the form ∑_{k} ε_{k} □_{q−1}^{k} for some ε_{k} ∈ {−1, +1}. Here, in this sum, we write only those (q − 1)–cubes which appear with non–zero coefficient. One sees that ∂ ∘ ∂ = 0, but, obviously, the homology of the chain complex (C_{∗}, ∂) is trivial: it is just the homology of ℝ^{s}. In order to get a more interesting (co)homology, one needs to consider a weight function w : Q_{q} → ℤ (0 ≤ q ≤ s). In the present case this will be defined, for each k ∈ Char fixed, by

w(□_{q}) := max{χ_{k}(l) : l is a vertex of □_{q}}.

Once the weight function is defined, one considers F^{q}, the set of morphisms Hom_{ℤ}(C_{q}, T_{0}^{+}) with finite support on Q_{q}. Notice that F^{q} is, in fact, a ℤ[U]–module by (p ∗ ϕ)(□_{q}) := p(ϕ(□_{q})) (ϕ ∈ F^{q}, p ∈ ℤ[U]). Moreover, F^{q} has a 2ℤ–grading: ϕ ∈ F^{q} is homogeneous of degree 2d ∈ ℤ if for each □_{q} ∈ Q_{q} with ϕ(□_{q}) ≠ 0, ϕ(□_{q}) is a homogeneous element of T_{0}^{+} of degree 2d − 2 · w(□_{q}).

Next, one defines δ : F^{q} → F^{q+1}. For this, fix ϕ ∈ F^{q} and we show how δ(ϕ) acts on a cube □_{q+1} ∈ Q_{q+1}. First write ∂□_{q+1} = ∑_{k} ε_{k}□_{q}^{k}, then set

(2.2.1)  (δ(ϕ))(□_{q+1}) := ∑_{k} ε_{k} U^{w(□_{q+1})−w(□_{q}^{k})} ϕ(□_{q}^{k}).

One verifies that δ ∘ δ = 0, i.e. (F^{∗}, δ) is a cochain complex (with δ homogeneous of degree zero); its homology is denoted by {ℍ^{q}(Γ, k)}_{q≥0}.

(F^{∗}, δ_{w}) has a natural augmentation too. Indeed, set m_{k} := min_{l∈L}{χ_{k}(l)}. Then one defines the ℤ[U]–linear map ϵ : T_{2m_{k}}^{+} ⟶ F^{0} such that ϵ(U^{−m_{k}−s})(l) is the class of U^{−m_{k}+χ_{k}(l)−s} in T_{0}^{+} for any integer s ≥ 0. Then ϵ is injective and homogeneous of degree zero, δ ∘ ϵ = 0. The homology of the augmented cochain complex

0 ⟶ T_{2m_{k}}^{+} ⟶^{ϵ} F^{0} ⟶^{δ} F^{1} ⟶^{δ} …

is called the reduced lattice cohomology ℍ_{red}^{∗}(Γ, k). For any q ≥ 0, both ℍ^{q} and ℍ_{red}^{q} admit an induced graded ℤ[U]–module structure and ℍ^{q} = ℍ_{red}^{q} for q > 0.

Note that ϵ provides a canonical embedding of T_{2m_{k}}^{+} into ℍ^{0}. Moreover, one has a graded ℤ[U]–module isomorphism ℍ^{0} = T_{2m_{k}}^{+} ⊕ ℍ_{red}^{0}, and ℍ_{red}^{∗} has finite ℤ–rank.

Remark 2.2.2. For the definition of the lattice cohomology for more general weight functions and graphs with non–zero genera, see [N08].

2.3. Reinterpretation of the lattice cohomology. If k′ = k + 2l for some l ∈ L then ℍ^{∗}(Γ, k) and ℍ^{∗}(Γ, k′) are isomorphic up to a degree shift, cf. [N08, (3.3)]. In fact, all the modules {ℍ^{∗}(Γ, k)}_{k} can be packed into only one object, more in the spirit of [OSz03]. This was used in [Gr08] too.

In this way, for any fixed k ∈ Char, L is identified with the sublattice k + 2L ∈ Char, and with the notation l′ := k + 2l one has χ_{k}(l) = −\frac{1}{8}(l′, l′) + \frac{1}{8}(k, k). In particular, up to a shift in degree, for each fixed k, l ↦ χ_{k}(l) and l′ ↦ −\frac{1}{8}(l′, l′) define the same weight function (on the cubes, see below), hence the same cohomology. In fact, we will even modify this weight function by s/8 (in this way the blowing up will induce a degree preserving isomorphism, cf. §3), and set:

(2.3.1)  w : Char → ℚ,  w(k) := −\frac{k^{2} + s}{8} (s = |J |).

The q–cubes □_{q} ∈ Q_{q} are associated with pairs (k, I) ∈ Char × P(J), |I| = q (here P(J) denotes the power set of J), and have the form {k + 2 ∑_{j∈I′} E_{j})_{I′}, where I′ runs over all subsets of I. The weights are defined by

(2.3.2)  w(□_{q}) = w((k, I)) = max_{I′⊂I} { w(k + 2 ∑_{j∈I′} E_{j}) }.

Moreover, F^{q} are elements of Hom_{ℤ}(Q_{q}, T_{0}^{+}) with finite support. Similarly as above, F^{q} is a ℤ[U]–module with a ℚ–grading: ϕ ∈ F^{q} is homogeneous of degree r if for each □_{q} ∈ Q_{q} with ϕ(□_{q}) ≠ 0, ϕ(□_{q}) is a homogeneous element of T_{0}^{+} of degree r − 2 · w(□_{q}).

It is convenient to consider the module of (infinitely supported) homological cycles too: let F_{q} be the direct product of ℤ_{≥0}×Q_{q} copies of ℤ (considered already in [OSz03] for q = 0). We write the pair (m, □) as U^{m}□. F_{q} becomes a ℤ[U]–module by U(U^{m}□) = U^{m+1}□. Clearly F^{q} = Hom_{ℤ[U]}(F_{q}, T_{0}^{+}), i.e. ϕ(U□) = Uϕ(□) for any ϕ.

δ : F^{q} → F^{q+1} is defined as in (2.2.1) using the new weight function, or by δ(ϕ)(□) = ϕ(∂(□)), where for □ = (k, I) = (k, {j_{1}, …, j_{q}}) one has (2.3.3)

∂(k, I) = ∑_{l=1}^{q}(−1)^{l}( U^{w(k,I)−w(k,I\j_{l})}(k, I\j_{l})−U^{w(k,I)−w(k+2E_{j_{l}} ,I\j_{l})}(k+2E_{j_{l}}, I\j_{l}) ).

The cohomology of (F^{∗}, δ) is denoted by ℍ^{∗}(Γ). Since the vertices of a cube belong to the same class Char/2L (where a class has the form [k] = {k + 2l}_{l∈L} ⊂ Char), ℍ^{∗}(Γ) has a natural direct sum decomposition

ℍ^{∗}(Γ) = ⊕_{[k]∈Char/2L} ℍ^{∗}(Γ, [k]).

In fact, if [k_{1}] = [k_{2}] then w(k_{1}) − w(k_{2}) ∈ ℤ.

Since Γ is negative definite, for each class [k] ∈ Char/2L one has a well–defined rational number

d[k] := − max_{k∈[k]} \frac{k^{2} + s}{4} = 2 · min_{k∈[k]} w(k).

Then (cf. 2.2) one has a direct sum decomposition:

(2.3.4)  ℍ^{0}(Γ, [k]) = T_{d[k]}^{+} ⊕ ℍ_{red}^{0}(Γ, [k]).

Sometimes we write 𝕋 := ⊕_{[k]}T_{d[k]}^{+} for the canonical submodule of ℍ^{0}, which satisfies ℍ^{0} = 𝕋 ⊕ ℍ_{red}^{0}.

Following [N05, N08, N10] we define the Euler characteristic of ℍ^{∗}(Γ, [k]) by

(2.3.5)  eu(ℍ^{∗}(Γ, [k])) := −d[k]/2 + ∑_{q≥0}(−1)^{q} rank_{ℤ} ℍ_{red}^{q}(Γ, [k]).

Remark 2.3.6. For any ϕ ∈ F^{q} and ℓ ≥ 0 set U^{−ℓ} ∗ ϕ ∈ F^{q} defined as follows: if ϕ(□) = ∑_{m≥0} a_{m,□}U^{−m}, then (U^{−ℓ} ∗ ϕ)(□) = ∑_{m≥0} a_{m,□}U^{−m−ℓ}. Notice that U(U^{−1} ∗ ϕ) = ϕ (but, in general, U^{−1} ∗ (Uϕ) ≠ ϕ).

For any □ ∈ Q_{q} let □^{∨} be that element of F^{q} which sends □ to 1 ∈ T_{0}^{+} and any other element to zero. Then any element of F^{q} is a finite ℤ–linear combination of elements of type U^{−ℓ} ∗ □^{∨}.

2.4. Generalizations. Graphs which are not negative definite. Since the definition of the lattice cohomology is purely algebraic/combinatorial, its definition can be considered for any graph, or even for any lattice L with fixed basis elements {E_{j}}_{j}. Nevertheless, in this note we assume that the graphs are non– degenerate. Of course, doing this generalization, some of the properties of the lattice cohomology associated with connected negative definite graphs will not survive. Here we wish to point out some of the differences.

First, we notice that dropping the connectedness of the graph basically has no effect (this fact already was used in the main inductive proofs of [OSz03] and [N05, (8.3)]). Dropping the negative definiteness is more serious. In order to explain this, we will introduce the following spaces: for any fixed class [k] and any real number r we define S_{r} as the union of all cubes □ ∈ Q_{q} with all vertices in [k] and w(□) ≤ r.

If Γ is negative definite then S_{r} is compact for any r. In particular, the weight function takes its minimum on it, and for each class [k] = Char/2L a decomposition ℍ^{0}(Γ, [k]) = T_{d([k])}^{+} ⊕ ℍ_{red}^{0}(Γ, [k]) can be defined. This property will not survive for general graphs. In fact, if the components of S_{r} are not compact, then we even might have the vanishing ℍ^{0}(Γ) = 0 (and this can happen simultaneously with the non–vanishing of ℍ^{1}), see e.g. (2.4.1). Also, in [N08, Theorem 3.1.12] the lattice cohomology is recovered from the simplicial cohomology of the spaces S_{r}. In the general case, the adapted version of that theorem (with similar proof), valid for any lattice, states that the same formula is valid once if we replace the cohomology groups H^{q}(S_{r}, ℤ) by the cohomology groups with compact support H_{c}^{q}(S_{r}, ℤ). Namely

ℍ^{∗}(Γ, [k]) ≃ ⨁_{r∈w(k)+ℤ} H_{c}^{∗}(S_{r}, ℤ).

We wish to emphasize that replacing the cohomology with cohomology with compact support is a conceptual modification: the two groups have different functorial properties.

In fact, exactly this second point of view is crucial in the definition of the lattice cohomology (and in the definition of the maps in the surgery formula treated next): basically we mimic the infinitely supported homology and the (dual) cohomology with compact support, and the corresponding functors associated with them.

Since the surgery exact sequence considered in the next section is valid for any non–degenerate lattice, it is very convenient to extend the theory to arbitrary graphs (or at least for non–degenerate ones) in order to have a larger flexibility for computations. Nevertheless, we have to face the following crucial problem. An oriented plumbed 3–manifold M(Γ) has many different plumbing representations Γ. They are connected by the moves of the (oriented) plumbing calculus. For negative definite graphs the only moves are the blowups (and their inverses) by (rational) (−1)-vertices. In [N08, (3.4)] it is proved that the lattice cohomology is stable with respect to these moves, see also §3 here. On the other hand, if we enlarge our plumbing graphs, this stability condition will not survive: the same 3–manifold can be represented by many different lattices with rather different lattice cohomologies.

More precisely, for negative definite graphs one conjectures (cf. [N08]) that from the lattice cohomology one can recover (in a combinatorial way) the Heegaard– Floer homology of Ozsváth–Szabó. In particular, the lattice cohomology carries a geometric meaning depending only on M(Γ). This geometric meaning is lost in the context of general graphs (or, at least, it is not so direct).

Example 2.4.1. S^{3} can be represented by a graph with one vertex, which has decoration −1. Computing the lattice cohomology of this graph we get ℍ^{q} = 0 for q ≠ 0 and ℍ^{0} = ℍ_{red}^{0} = T_{0}^{+}. (This is the Heegaard–Floer homology HF^{+}(S^{3}).) On the other hand, it is instructive to compute the lattice cohomology of another graph, which has one vertex, but now with decoration +1. Its lattice cohomology is ℍ^{q} = 0 for q ≠ 1 and ℍ^{1} = T_{−1/2}^{+}. This graph also represents S^{3}, and the two graphs can be connected by a sequence of non–empty graphs and ±1–blowups and blowdowns. In particular, we conclude that the lattice cohomology is not stable with respect to blowing up/down (+1)–vertices.

Remark 2.4.2. Replacing negative definite graphs with arbitrary non–degenerate ones we can also adopt the definition of the weight function (2.3.1) by modifying to (2.4.3)

w_{†} : Char → ℚ, w_{†}(k) := \frac{−k^{2} + 3σ + 2s}{8} (σ = signature of ( , ), s = |J |),

a more common expression associated with (4–manifold intersection) forms which are not negative definite. In this note we will not do this, but the interested reader preferring this expression might shift all the weights below by 3(σ + s)/8.

3. Blowing up Γ

3.1. Since in the main construction we will need some of the operators induced by blowing down, we will make explicit the involved morphisms.

The next discussion provides a new proof of the stability of the lattice cohomology in the case when we blow up a vertex (for the old proof see [N08, (3.4)]).

Starting from now, the graph is neither necessarily connected nor necessarily negative definite. Nevertheless, we will assume that the intersection form is non– degenerate.

We assume that Γ′ is obtained from Γ by ‘blowing up the vertex j_{0}’. More precisely, Γ′ denotes a graph with one more vertex and one more edge: we glue to the vertex j_{0} by the new edge the new vertex E_{new} with decoration −1 (and genus 0), while the decoration of E_{j_{0}} is modified from e_{j_{0}} into e_{j_{0}} − 1, and we keep all the other decorations. We will use the notations L(Γ), L(Γ′), L′(Γ), L′(Γ′). Let w′ : Char(Γ′) → ℚ be the weight function of Γ′ defined similarly as in (2.3.1). (We may use the following convention for the ordering of the indices: if j ≠ j_{0}, then j < j_{0} < j_{new}.) The following facts can be verified:

3.1.1. Consider the maps π_{∗} : L(Γ′) → L(Γ) defined by

π_{∗}(∑ x_{j}E_{j} + x_{new}E_{new}) = ∑ x_{j}E_{j},

and π^{∗} : L(Γ) → L(Γ′) defined by

π^{∗}(∑ x_{j}E_{j}) = ∑ x_{j}E_{j} + x_{j_{0}}E_{new}.

Then (π^{∗}x, x′)_{Γ′} = (x, π_{∗}x′)_{Γ}. This shows that

(π^{∗}x, π^{∗}y)_{Γ′} = (x, y)_{Γ} and (π^{∗}x, E_{new})_{Γ′} = 0

for any x, y ∈ L(Γ). Both π_{∗} and π^{∗} extend over L ⊗ ℚ and L′.

3.1.2. Define the (nonlinear) map: c : L′(Γ) → L′(Γ′), c(l′) := π^{∗}(l′) + E_{new}. Then c(Char(Γ)) ⊂ Char(Γ′) and c induces an isomorphism between the orbit spaces Char(Γ)/2L(Γ) ≃ Char(Γ′)/2L(Γ′). Moreover, for any k ∈ Char(Γ) and k′ := c(k) one has w′(k′) = w(k).

3.1.3. π_{∗}(Char(Γ′)) ⊂ Char(Γ). For any k′ ∈ Char(Γ′) write k := π_{∗}(k′). Then k′ = π^{∗}(k) + aE_{new} for some odd integer a, and w′(k′) − w(k) = \frac{a^{2}−1}{8} ∈ ℤ_{≥0}.

The maps c and π_{∗} can be extended to the level of cubes and complexes as follows.

3.1.4. For any q ≥ 0 and □ = (k′, I) ∈ Q_{q}(Γ′) one defines π_{∗}((k′, I)) := (π_{∗}(k′), I) ∈ Q_{q}(Γ), provided that j_{new} ∉ I. By (3.1.3) one has w_{Γ′}(□)−w_{Γ}(π_{∗}(□)) ≥ 0. This defines a homological morphism π_{∗}^{h} : F_{q}(Γ′) → F_{q}(Γ) by

(3.1.5)  π_{∗}^{h}(□) = \{ \begin{matrix}U^{w_{Γ′} (□)−w_{Γ}(π_{∗}(□))} π_{∗}(□) & if j_{new} ∉ I, \\ 0 & else. \end{matrix}

Using (2.3.3) one verifies that π_{∗}^{h} ∘ ∂ = ∂ ∘ π_{∗}^{h}, hence π_{∗}^{h} is morphism of homological complexes. In particular, its dual π_{∗}^{c} : F^{q}(Γ) → F^{q}(Γ′), defined by π_{∗}^{c}(ϕ) = ϕ ∘ π_{∗}^{h}, satisfies π_{∗}^{c} ∘ δ = δ ∘ π_{∗}^{c}, hence it is a morphism of complexes as well.

3.1.6. Before we extend c to the level of complexes, notice that for any k ∈ Char(Γ)

c(k + 2E_{j}) = \{ \begin{matrix}c(k) + 2E_{j} & if j ≠ j_{0}, \\ c(k) + 2E_{j} + 2E_{new} & if j = j_{0}. \end{matrix}

In particular, the pair c(k) and c(k+2E_{j_{0}}) does not form a 1–cube in Γ′. Hence, the mapping (k, I) ↦ (c(k), I), sending the vertex k+2 ∑_{j∈I′} E_{j} into c(k)+2 ∑_{j∈I′} E_{j}, does not commute with the boundary operator. The ‘right’ operator c^{h} : F_{q}(Γ) → F_{q}(Γ′) associates with c^{h}((k, I))

\{ \begin{matrix}(c(k), I) & if j_{0} ∉ I, \\ (c(k), I) + U^{w(k,I)−w(k+2E_{j_{0}} ,I_{0})}(c(k) + 2E_{j_{0}}, I_{0} ∪ j_{new}) & if I = I_{0} ∪ j_{0}, j_{0} ∉ I_{0}. \end{matrix}

In fact, by (3.1.7) below, w(k + 2E_{j_{0}}, I_{0}) above can be replaced by

w′(c(k) + 2E_{j_{0}}, I_{0}), or even by w′(c(k) + 2E_{j_{0}}, I_{0} ∪ j_{new}).

By (3.1.2) and a computation one gets (where in the third line I′_{0} is any subset of J with j_{0} ∉ I′_{0}): (3.1.7)

\begin{matrix}w((k, I)) = & w′((c(k), I)), \\ w((k + 2E_{j_{0}}, I_{0})) = & w′((c(k) + 2E_{j_{0}}, I_{0})), \\ w′(c(k) + 2 ∑_{j∈I′_{0}} E_{j} + 2E_{j_{0}}) = & w′(c(k) + 2 ∑_{j∈I′_{0}} E_{j} + 2E_{j_{0}} + 2E_{new}). \end{matrix}

These and a (longer) computation shows that c^{h} commutes with the boundary operator ∂.

3.1.8. Using π_{∗} ∘ c, the definitions and the first equation of (3.1.7) one gets π_{∗}^{h} ∘c^{h} = id_{F_{∗}(Γ)}. On the other hand, c^{h} ∘π_{∗}^{h} is not the identity, but it is homotopic to id_{F_{∗}(Γ′)}. Indeed, we define the homotopy operator K : F_{∗}(Γ′) → F_{∗+1}(Γ′) as follows. Write any k as cπ_{∗}(k) + 2aE_{new} for some a ∈ ℤ. Then define K((k, I)) as 0 if either j_{new} ∈ I or a = 0. Otherwise take for K((k, I))

sign(a) · ∑ U^{w(k,I)−w(cπ_{∗}(k)+2lE_{new},I∪j_{new})} (cπ_{∗}(k) + 2lE_{new}, I ∪ j_{new}),

where the summation is over l ∈ {0, 1, …, a − 1} if a > 0 and l ∈ {a, …, −1} if a < 0. (The exponents are non–negative because of (3.1.3).) Then, again by a computation, ∂ ∘ K − K ∘ ∂ = id − c^{h} ∘ π_{∗}^{h}.

In particular, π_{∗}^{h} and c^{h} induce (degree preserving) isomorphisms of the corresponding lattice cohomologies. In the sequel the operator π_{∗}^{h} will be crucial.

4. Comparing Γ and Γ \ j_{0}

4.1. Notations, remarks. We consider a non–degenerate graph Γ as in 3.1, and we fix one of its vertices j_{0} ∈ J. The new graph Γ \ j_{0} is obtained from Γ by deleting the vertex j_{0} and all its adgacent edges. We will denote by L(Γ), L′(Γ), L(Γ\ j_{0}), L′(Γ \ j_{0}) the corresponding lattices.

The operator i : L(Γ \ j_{0}) → L(Γ), i(E_{j,Γ\j_{0}}) = E_{j,Γ} identifies L(Γ \ j_{0}) with a sublattice of L(Γ). The dual operator (restriction) is R : L′(Γ) → L′(Γ \ j_{0}), R(E_{j,Γ}^{∗}) = E_{j,Γ\j_{0}}^{∗} for j ≠ j_{0} and R(E_{j_{0},Γ}^{∗}) = 0. It satisfies (i_{ℚ}(x), y)_{Γ} = (x, R(y))_{Γ\j_{0}} for any x ∈ L′(Γ \ j_{0}) and y ∈ L′(Γ). (Here, E_{j,Γ}^{∗} respectively E_{j,Γ\j_{0}}^{∗} are the usual dual generators of L′ considered in Γ, respectively in Γ \ j_{0}.)

Recall that l′ = ∑_{j} a_{j}E_{j,Γ}^{∗} is characteristic if and only if a_{j} ≡ e_{j} (mod 2). In particular, R sends characteristic elements into characteristic elements. On the other hand, i_{ℚ} does not necessarily preserve characteristic elements.

Although R(Char(Γ)) ⊂ Char(Γ \ j_{0}), this operator cannot be extended to the level of cubes. Indeed, notice that

(4.1.1)  R(E_{j_{0}}) = − ∑_{(j,j_{0})∈E_{Γ}} E_{j,Γ\j_{0}}^{∗},

where the sum runs over the adjacent vertices of j_{0} in Γ. In particular, the endpoints of the 1–cube (k, j_{0}) are sent into R(k) and R(k) + 2R(E_{j_{0}}), which cannot be expressed as combinations of 1–cubes in Γ \ j_{0}. In the next subsection we will consider another operator, which can be extended to the level of cubes (and which, in fact, operates in a different direction than R, cf. 2.4).

4.2. The B–operator. Consider b : L′(Γ \ j_{0}) → L′(Γ) defined by

∑_{j} a_{j}E_{j,Γ\j_{0}}^{∗} ↦ ∑_{j} a_{j}E_{j,Γ}^{∗}.

Clearly, if k ∈ Char(Γ \ j_{0}) then b(k) + a_{j_{0}}E_{j_{0},Γ}^{∗} ∈ Char(Γ) for any a_{j_{0}} with a_{j_{0}} ≡ e_{j_{0}} (mod 2). In order to see how it operates on cubes, notice that in Γ \ j_{0} for any j ∈ J (Γ \ j_{0}) one has

E_{j,Γ\j_{0}} = −e_{j} E_{j,Γ\j_{0}}^{∗} − ∑_{(i,j)∈E_{Γ\j_{0}}} E_{i,Γ\j_{0}}^{∗},

and a similar relation holds in Γ too. Therefore, one gets that

(4.2.1)  b(E_{j}) = E_{j} + (E_{j_{0}}, E_{j})_{Γ} · E_{j_{0},Γ}^{∗}.

Therefore,

(4.2.2)  b(l′) = i_{ℚ}(l′) + (E_{j_{0}}, i_{ℚ}(l′))_{Γ} · E_{j_{0},Γ}^{∗}.

In particular, b is a ‘small’ modification of i_{ℚ}, but this modification is enough to extend it to the level of cubes. Although the vertices of (k, I) are not sent to the vertices of a cube (provided that I contains elements adjacent to j_{0} in Γ), cf. (4.2.1), nevertheless if we consider all the shifts in the direction of the ‘error’ of (4.2.1) we get a well–defined operator

B_{q} : F_{q}(Γ \ j_{0}) → F_{q}(Γ)

given by

(4.2.3)  B_{q}((k, I)) := ∑_{a_{j_{0}} ≡e_{j_{0}} (mod 2)} (b(k) + a_{j_{0}}E_{j_{0},Γ}^{∗}, I).

(Here we keep the notation ‘B’, since this notation was used in similar contexts, as in [OSz03, N05, Gr08].)

B_{q} induces a morphism B^{q} : F^{q}(Γ) → F^{q}(Γ \ j_{0}) via (B^{q}ϕ)(□) = ϕ(B_{q}□). A straightforward (slightly long) computation shows that B_{q} commutes with the boundary operator ∂, hence B^{∗} ∘ δ = δ ∘ B^{∗} too. In particular, one gets a well– defined morphism of ℤ[U]–modules 𝔹^{∗} : ℍ^{∗}(Γ) → ℍ^{∗}(Γ \ j_{0}).

4.3. The sign–modified B–operator. In the exact sequences considered in the next section we will need to modify the B–operator by a sign (compare with the end of §2 of [OSz03], where the case q = 0 is discussed). The definition depends on some choices.

For each L–orbit [k] := k+2L(Γ\j_{0}) ⊂ Char(Γ\j_{0}) we will fix a representative r_{[k]} ∈ [k]. Hence, for any characteristic element k, with orbit [k], k−r_{[k]} ∈ 2L(Γ\j_{0}), hence (k − r_{[k]}, E_{j,Γ\j_{0}}^{∗}) ∈ 2ℤ for any j ≠ j_{0}. In particular, (k − r_{[k]}, R(E_{j_{0}})) ∈ 2ℤ for R(E_{j_{0}}) defined in (4.1.1). Then the modified operator

\overline{B}_{q} : F_{q}(Γ \ j_{0}) → F_{q}(Γ)

is given by

(4.3.1)  \overline{B}_{q}((k, I)) := ∑_{a_{j_{0}} ≡e_{j_{0}} (mod 2)} (−1)^{n(k,a_{j_{0}} )/2} · (b(k) + a_{j_{0}}E_{j_{0},Γ}^{∗}, I),

where

n(k, a_{j_{0}}) := (k − r_{[k]}, R(E_{j_{0}})) + a_{j_{0}} + e_{j_{0}}.

Then, again, \overline{B}^{∗} commutes with the boundary operator, hence \overline{B}^{∗} : F^{q}(Γ) → F^{q}(Γ \ j_{0}) defined by (\overline{B}^{q}ϕ)(□) = ϕ(\overline{B}_{q}□) commutes with δ, hence it also defines a ℤ[U]–module morphism \overline{𝔹}^{∗} : ℍ^{∗}(Γ) → ℍ^{∗}(Γ \ j_{0}).

Remark 4.3.2. The morphisms 𝔹^{∗} and \overline{𝔹}^{∗} do not preserve neither the gradings nor the orbits Char/2L (i.e. they do not split into a direct sum with respect to these orbits).

5. The surgery exact sequence

5.1. Notations. Let Γ be a non–degenerate graph as above with s = |J | ≥ 2. We fix a vertex j_{0} ∈ J. Associated with j_{0} we consider two other graphs, namely Γ \ j_{0} and Γ_{j_{0}}^{+}. The second one is obtained from Γ by modifying the decoration e_{j_{0}} of the vertex j_{0} into e_{j_{0}} + 1. In order to stay in our category of objects, we will assume that both graphs Γ \ j_{0} and Γ_{j_{0}}^{+} are non–degenerate.

Our goal is to establish a long exact sequence connecting the lattice cohomologies of these three graphs, similar to one which is valid for the Heegaard–Floer cohomologies provided by surgeries. As usual, in order to define a long exact sequence, we need first to determine a short exact sequence of complexes.

The graphs Γ and Γ\j_{0} will be connected by the \overline{B}–operator, cf. 4.3. We define the ‘A’–operator connecting Γ and Γ_{j_{0}}^{+} as follows. Let Γ^{b} be the graph obtained from Γ by attaching a new vertex j_{new} (via a single new edge) to j_{0}, such that the decoration of the new vertex is −1. Then, for the pair Γ, Γ^{b} (since Γ = Γ^{b} \ j_{new}), we can apply the construction and results of §4. In particular, we get morphisms of complexes: B_{∗} : F_{∗}(Γ) → F_{∗}(Γ^{b}) and B^{∗} : F^{∗}(Γ^{b}) → F^{∗}(Γ). Next, since Γ_{j_{0}}^{+} is obtained from Γ^{b} by blowing down the new vertex j_{new}, by § 3 we get morphisms of complexes: π_{∗}^{h} : F_{∗}(Γ^{b}) → F_{∗}(Γ_{j_{0}}^{+}) and π_{∗}^{c} : F^{∗}(Γ_{j_{0}}^{+}) → F^{∗}(Γ^{b}). By composition we get the A–operators

A_{∗} := π_{∗}^{h} ∘ B_{∗} : F_{∗}(Γ) → F_{∗}(Γ_{j_{0}}^{+}), and A^{∗} := B^{∗} ∘ π_{∗}^{c} : F^{∗}(Γ_{j_{0}}^{+}) → F^{∗}(Γ).

In particular, we can consider the short sequence of complexes

(5.1.1)  0 → F^{∗}(Γ_{j_{0}}^{+}) ⟶^{A^{∗}} F^{∗}(Γ) ⟶^{\overline{B}^{∗}} F^{∗}(Γ \ j_{0}) → 0.

Theorem 5.1.2. The short sequence of complexes (5.1.1) is exact.

The proof is given in several steps.

5.2. The injectivity of A^{q}. Take an arbitrary non–zero ϕ ∈ F^{q}(Γ_{j_{0}}^{+}). In order to prove that A^{q}ϕ ≠ 0, we need to find f ∈ F_{q}(Γ) such that ϕ(A_{q}(f)) ≠ 0. Let N be the smallest non–negative integer such that U^{N+1}ϕ = 0. Then replacing ϕ by U^{N}ϕ we may assume that Uϕ = 0. In particular, in f (or in A_{q}(f)) any term whose coefficient has the form U^{n} (n > 0) is irrelevant.

Since ϕ ≠ 0, there exists (k̄, I) ∈ Q_{q}(Γ_{j_{0}}^{+}) with ϕ((k̄, I)) ≠ 0. Write

(5.2.1)  k̄ = ∑_{j∈J} a_{j}E_{j,Γ_{j_{0}}^{+}}^{∗} + E_{j_{0},Γ_{j_{0}}^{+}}^{∗} ,

with a_{j} − e_{j} even for all j ∈ J. Since ϕ is finitely supported, we may assume that

(5.2.2)  a_{j_{0}} is minimal with the property ϕ((k̄, I)) ≠ 0.

We wish to construct f ∈ F_{q}(Γ) such that ϕ(A_{q}(f) − (k̄, I)) = 0. For the weight functions in Γ^{b} and Γ_{j_{0}}^{+} we will use the notations w^{b}, respectively w^{+}.

First, set

(5.2.3)  k := ∑_{j} a_{j}E_{j,Γ}^{∗} ∈ Char(Γ).

Then

B_{0}(k) = ∑_{a≡1} k_{a}^{b}, where k_{a}^{b} = ∑_{j} a_{j}E_{j,Γ^{b}}^{∗} + aE_{new}^{∗}.

Set also

k_{a} := π_{∗}(k_{a}^{b}) = ∑_{j} a_{j}E_{j,Γ_{j_{0}}^{+}}^{∗} + aE_{j_{0},Γ_{j_{0}}^{+}}^{∗} .

Notice that k_{1} = k̄. Since π^{∗}k_{a} = k_{a}^{b} − aE_{new}, by (3.1.3)

(5.2.4)  w^{b}(k_{a}^{b}) = w^{+}(k_{a}) + (a^{2} − 1)/8.

Moreover, B((k, I)) = ∑_{a≡1}(k_{a}^{b}, I) and π_{∗}((k_{a}^{b}, I)) = (k_{a}, I).

For any I′ ⊂ I write E_{I′} = ∑_{j∈I′} E_{j}. Let δ_{I′} be 1 if j_{0} ∈ I′, and zero otherwise. Since

(5.2.5)  w^{b}(k_{a}^{b} + 2E_{I′}) − w^{b}(k_{a}^{b}) = \frac{1}{2} ∑_{j∈I′} a_{j} − \frac{1}{2}(E_{I′}, E_{I′})_{Γ^{b}},

and

(5.2.6)  w^{+}(k_{a} + 2E_{I′}) − w^{+}(k_{a}) = \frac{1}{2} ∑_{j∈I′} a_{j} + \frac{1}{2}δ_{I′}a − \frac{1}{2}(E_{I′}, E_{I′})_{Γ_{j_{0}}^{+}} ,

we get

(5.2.7)  w^{b}(k_{a}^{b} + 2E_{I′}) − w^{+}(k_{a} + 2E_{I′}) = \frac{a^{2} − 1}{8} − δ_{I′} · \frac{a − 1}{2}.

Assume that I ∌ j_{0}. Then, by (5.2.7), we get that w^{b}(k_{a}^{b}, I)−w^{+}(k_{a}, I) = (a^{2}−1)/8, hence

(5.2.8)  A_{q}((k, I)) = ∑_{a≡1, a<0} U^{\frac{a^{2}−1}{8}} (k_{a}, I) + (k̄, I) + ∑_{a≡1, a≥3} U^{\frac{a^{2}−1}{8}} (k_{a}, I).

Both sums are killed by ϕ, the first one by (5.2.2), the second one by Uϕ = 0, hence ϕ(A_{q}((k, I)) − (k̄, I)) = 0.

Next, assume that I ∋ j_{0}. In that case, again by (5.2.7), we get that w^{b}(k_{a}^{b}, I)− w^{+}(k_{a}, I) > 0 whenever a ∉ {−1, 1, 3}. For the other three special values we have the following facts. It is convenient to define for each k (represented as in (5.2.3)) the integer

(5.2.9)  M(k) := max_{I′} { ∑_{j∈I′} a_{j} − (E_{I′}, E_{I′})_{Γ^{b}} }.
• For a = 1 one has w^{b}(k_{a}^{b}, I) − w^{+}(k_{a}, I) = 0 always.

• If a = 3, then the right hand side of (5.2.7) is 1 − δ_{I′}, hence w^{b}(k_{a}^{b}, I) − w^{+}(k_{a}, I) = 0 if and only if max_{I′}{ w^{b}(k_{a}^{b} + 2E_{I′}) } can be realized by some I′ with I′ ∋ j_{0}. By (5.2.5) this is equivalent to

(5.2.10)  M(k) can be realized by some I′ with I′ ∋ j_{0}.

• If a = −1, then the right hand side of (5.2.7) is δ_{I′}, hence w^{b}(k_{a}^{b}, I) − w^{+}(k_{a}, I) = 0 if and only if

(5.2.11)  M(k) can be realized by some I′ with I′ ∌ j_{0}.

Assume that in the case I ∋ j_{0} for a = 3 one has w^{b}(k_{a}^{b}, I)−w^{+}(k_{a}, I) > 0. Then by an identical argument as in the case I ∌ j_{0} we get that ϕ(A_{q}((k, I))−(k̄, I)) = 0.

Hence, the remaining final case is when there exists at least one I′ ⊂ I which contains j_{0} and satisfies (5.2.10). Then

A_{q}((k, I) = (k̄, I) + (k̄ + 2E_{j_{0},Γ_{j_{0}}^{+}}^{∗} , I) (mod U and ker(ϕ)).

Now, we will consider k + 2E_{j_{0},Γ}^{∗} instead of k. Via the operator A_{q} it provides the same cubes as k (where the index set will have a shift a ↦ a − 2) but with different U^{m}–coefficients. Notice that by (5.2.10) and (5.2.11), M(k + 2E_{j_{0},Γ}^{∗}) can be realized only by subsets I′ with I′ ∋ j_{0}, hence (5.2.11) will fail. Therefore,

A_{q}((k + 2E_{j_{0},Γ}^{∗}, I) = (k̄ + 2E_{j_{0},Γ_{j_{0}}^{+}}^{∗} , I) + (k̄ + 4E_{j_{0},Γ_{j_{0}}^{+}}^{∗} , I) (mod U).

More generally, for any positive integer ℓ, by the same argument

A_{q}((k + 2ℓE_{j_{0},Γ}^{∗}, I) = (k̄ + 2ℓE_{j_{0},Γ_{j_{0}}^{+}}^{∗} , I) + (k̄ + (2ℓ + 2)E_{j_{0},Γ_{j_{0}}^{+}}^{∗} , I) (mod U).

Since ϕ is finitely supported, ϕ((2ℓ+2)E_{j_{0},Γ_{j_{0}}^{+}}^{∗} , I)) = 0 for some ℓ ≥ 0, let us consider

the minimal such ℓ. Then A_{q} modulo U and ker(ϕ) restricted to the relevant finite dimensional spaces looks like an (ℓ + 1) × (ℓ + 1) upper triangular matrix whose diagonal is the identity matrix. Since this is invertible over ℤ, the result follows: some linear combinations of the elements A_{q}((k + 2tE_{j_{0},Γ}^{∗}, I), 0 ≤ t ≤ ℓ, yield (k̄, I) modulo U and ker(ϕ).

5.3. The surjectivity of \overline{B}^{q}. We provide the same argument as [Gr08]: For any fixed a ≡ e_{j_{0}} (mod 2),

\overline{B}^{q}(U^{−ℓ} ∗ (b(k) + aE_{j_{0},Γ}^{∗}, I)^{∨}) = ±U^{−ℓ} ∗ (k, I)^{∨}.

Since the collection of U^{−ℓ} ∗ (k, I)^{∨} generates F^{q}(Γ \ j_{0}) (over ℤ), the surjectivity follows.

5.4. \overline{B}^{q} ∘ A^{q} = 0. Take an arbitrary (k, I) ∈ Q_{q}(Γ\j_{0}). Then, by (5.2.7), one has

(A_{q} ∘ \overline{B}_{q})((k, I)) = ∑_{a≡1} ∑_{c≡e_{j_{0}}} (−1)^{N(k)+ \frac{c+e_{j_{0}}}{2}} · U^{\frac{a^{2}−1}{8}} (π_{∗}b(k) + (a + c)E_{j_{0},Γ_{j_{0}}^{+}}^{∗} , I).

Those pairs (a, c) for which a^{2} − 1 and a + c are fixed hit the same element of F_{q}. Write a = 2i + 1. Then the two solutions of (a^{2} − 1)/8 = i(i + 1)/2 = t satisfy i_{1} + i_{2} = −1. Since the corresponding two c values satisfy 2i_{1} + c_{1} = 2i_{2} + c_{2}, one gets that (c_{2} − c_{1})/2 is odd. Hence the terms cancel each other in pairs.

5.5. ker \overline{B}^{q} ⊂ im A^{q}. Set ker U^{m} := \{ϕ ∈ F^{q}(Γ) : U^{m}ϕ = 0\}. Notice that the inclusion

(5.5.1)  ker U^{m} ∩ ker \overline{B}^{q} ⊂ im A^{q}

for m = 1 (together with (5.4)) implies by induction its validity for any m (cf. [Gr08]). Indeed, assume that the inclusion is true for m − 1 and set ϕ ∈ ker U^{m} ∩ ker \overline{B}^{q}. Then Uϕ = A^{q}(ψ) for some ψ. Moreover, ϕ̃:= ϕ−A^{q}(U^{−1} ∗ψ) ∈ ker U. On the other hand, by (5.4), ϕ̃ ∈ ker \overline{B}^{q} too. Therefore, by (5.5.1) applied for m = 1, we get ϕ̃ ∈ im A^{q}, hence ϕ ∈ im A^{q} too.

Notice that ∪_{m}(ker U^{m} ∩ ker \overline{B}^{q}) = ker \overline{B}^{q}, hence this would prove ker \overline{B}^{q} ⊂ im A^{q} too.

Next, we show (5.5.1) for m = 1. First notice that ker U ∩ ker \overline{B}^{q} is generated by elements of type (k, I)^{∨} +(k+2E_{j_{0},Γ}^{∗}, I)^{∨} where I ∌ j_{0}, and (k, I)^{∨} where I ∋ j_{0}.

In the first case (i.e. I ∌ j_{0}), let us write k as in (5.2.3), and set k̄ as in (5.2.1). Then by (5.2.7) and (5.2.8) one has

A_{q}((k, I)) = (k̄ − 2E_{j_{0},Γ_{j_{0}}^{+}}^{∗} , I) + (k̄, I) (mod U),

A_{q}((k + 2E_{j_{0},Γ}^{∗}, I)) = (k̄, I) + (k̄ + 2E_{j_{0},Γ_{j_{0}}^{+}}^{∗} , I) (mod U),

and (k̄, I) does not appear in any other term with non–zero coefficient (mod U). Hence A^{q}(k̄, I)^{∨} = (k, I)^{∨} + (k + 2E_{j_{0},Γ}^{∗}, I)^{∨}.

Next, we fix an element of type (k, I)^{∨} with I ∋ j_{0}. It belongs to the collection {(k(i), I)}_{a∈ℤ}, where k(i) = k+2iE_{j_{0},Γ}^{∗}, which will be treated simultaneously via the discussions of (5.2). Write k as in (5.2.3) and set k̄ via (5.2.1); it is also convenient to write k̄(i) := k̄ + 2iE_{j_{0},Γ_{j_{0}}^{+}}^{∗}. Notice that k(i) has coefficients {{a_{j}}_{j≠j_{0}}, a_{j_{0}} + 2i}.

Therefore, for i ≪ 0 (5.2.10) will fail and (5.2.11) is satisfied. Let i_{0} −1 be maximal when (5.2.10) fails. Then for i = i_{0} both conditions are satisfied, and for i > i_{0} only (5.2.10). Therefore, (5.5.2)

A_{q}((k(i), I)) = (k̄(i − 1), I) + (k̄(i), I)  (mod U) if i < i_{0},
A_{q}((k(i_{0}), I)) = (k̄(i_{0} − 1), I) + (k̄(i_{0}), I) + (k̄(i_{0} + 1), I) (mod U)
A_{q}((k(i), I)) = (k̄(i), I) + (k̄(i + 1), I)  (mod U) if i > i_{0}.

This reads as

A^{q}((k̄(i), I)^{∨}) = \{ \begin{matrix}(k(i), I)^{∨} + (k(i + 1), I)^{∨} & if i < i_{0} \\ (k(i), I)^{∨} & if i = i_{0} \\ (k(i), I)^{∨} + (k(i − 1), I)^{∨} & if i > i_{0} \end{matrix}

Taking finite linear combinations we get that any (k(i), I)^{∨} is in the image of A^{q}. This ends the proof of Theorem (5.1.2). As a corollary we get:

Theorem 5.5.3. Assume that the graphs Γ_{j_{0}}^{+}, Γ and Γ\j_{0} are non–degenerate. Then

⋯ ⟶ ℍ^{q}(Γ_{j_{0}}^{+}) ⟶^{𝔸^{q}} ℍ^{q}(Γ) ⟶^{\overline{𝔹}^{q}} ℍ^{q}(Γ \ j_{0}) ⟶^{ℂ^{q}} ℍ^{q+1}(Γ_{j_{0}}^{+}) ⟶ ⋯

is an exact sequence of ℤ[U]–modules.

6. The exact sequence for negative definite graphs

6.1. Preliminaries. For any graph Γ we write det(Γ) for the determinant of the negative of the intersection form associated with Γ. If Γ is negative definite then det(Γ) is obviously positive.

If Γ is negative definite then Γ \ j_{0} is automatically so for any j_{0}. Nevertheless, this is not the case for Γ_{j_{0}}^{+} (although, if Γ_{j_{0}}^{+} is negative definite then Γ is so too).

Lemma 6.1.1. Assume that Γ is negative definite. Then Γ_{j_{0}}^{+} is negative definite if and only if det(Γ) > det(Γ\j_{0}). If these conditions are satisfied then E_{j_{0},Γ}^{∗} ∉ L(Γ) and (E_{j_{0},Γ}^{∗})^{2} ∉ ℤ.

Proof. Γ_{j_{0}}^{+} is negative definite if and only if det(Γ_{j_{0}}^{+}) is positive (provided that Γ \ j_{0} is negative definite). But det(Γ_{j_{0}}^{+}) = det(Γ) − det(Γ \ j_{0}). The last statement follows from −(E_{j_{0},Γ}^{∗})^{2} = det(Γ \ j_{0})/ det(Γ), which cannot be an integer. □

In the sequel we assume that all the graphs are negative definite, but not necessarily connected. The next theorem is an addendum to Theorem 5.5.3. Since its proof is rather long, it will be published elsewhere.

Theorem 6.1.2. Assume that Γ_{j_{0}}^{+} is negative definite. Consider the exact sequence

0 ⟶ ℍ^{0}(Γ_{j_{0}}^{+}) ⟶^{𝔸^{0}} ℍ^{0}(Γ) ⟶^{\overline{𝔹}^{0}} ℍ^{0}(Γ \ j_{0}) ⟶^{ℂ^{0}} ℍ^{1}(Γ_{j_{0}}^{+}) ⟶ ⋯

and the canonical submodule 𝕋(Γ \ j_{0}) of ℍ^{0}(Γ \ j_{0}). Then 𝕋(Γ \ j_{0}) ⊂ im \overline{𝔹}^{0}; or, equivalently, the restriction ℂ^{0}|𝕋(Γ \ j_{0}) is zero.

6.2. Graphs with n bad vertices. We say that a negative definite connected graph is ‘rational’ if it is the plumbing representation of a the link of a rational singularity (or, the resolution graph of a rational singularity). They were characterized combinatorially by Artin; for more details see [N99, N05].

We fix an integer n ≥ 0. We say that a negative definite graph has at most n ‘bad’ vertices if we can find n vertices {j_{k}}_{1≤k≤n}, such that replacing their decorations e_{j_{k}} by some more negative integers e′_{j_{k}} ≤ e_{j_{0}} we get a graph whose all connected components are rational. (Notice that this is a generalization of the notion of ‘bad’ vertices of [OSz03]. A graph with at most one bad vertex is called ‘almost rational’ in [N05, N08]. Any ‘star–shaped’ graph, i.e. normal form of a Seifert manifold, has at most one bed vertex, namely the ‘central’ vertex.)

Theorem 6.2.1. If Γ has at most n bad vertices then ℍ_{red}^{q}(Γ) = 0 for q ≥ n.

This is a generalization of [N08, (4.3.3)], where it is proved for n = 1 (compare also with the vanishing theorems of [OSz03, N05]).

Proof. We run induction over n. If n = 0, then all the components of Γ are rational. By [N05], their reduced lattice cohomology vanishes. This fact remains true for more components too, since the cohomology of a tensor product of two acyclic complexes is acyclic.

Assume now that the statement is true for n−1 and take Γ with n bad vertices. Let j be one of them. Let Γ_{j}(−ℓ) be the graph obtained from Γ by replacing the decoration e_{j} by e_{j} − ℓ (ℓ ≥ 0). Then consider the long exact sequence (5.5.3) associated with Γ_{j}(−ℓ), Γ_{j}(−ℓ − 1) and Γ \ j_{0}, for all ℓ ≥ 0. Then, by the inductive step, we get that ℍ^{q}(Γ) = ℍ^{q}(Γ_{j}(−ℓ)) for all ℓ and q ≥ n. (Here, in the case n = 1, Theorem 6.1.2 is also used.) Since for ℓ ≫ 0 the graph Γ_{j}(−ℓ) has only n − 1 bad vertices, all these modules vanish. □

In fact, the above statement can be improved as follows.

Theorem 6.2.2. Assume that Γ has at most n ≥ 2 bad vertices {j_{k}}_{1≤k≤n} such that Γ \ j_{1} has at most (n − 2) bad vertices. Then ℍ_{red}^{q}(Γ) = 0 for q ≥ n − 1.

Proof. The proof is same as above, if one eliminates first the vertex j_{1}. □

See [N05, (8.2)(5.b)] for a graph Γ with 2 bad vertices {j_{1}, j_{2}} such that Γ \ j_{1} has only rational components.

7. The ‘relative’ surgery exact sequence

7.1. Preliminaries. The motivation for the next exact sequence is two–fold. First, the exact sequence (5.5.3) mixes the classes of the characteristic elements. (Note that the surgery exact sequence valid for the Heegaard–Floer theory — which is one of out models for the theory — does the same.) These classes, in topological language, correspond to the spin^{c}–structures of the corresponding plumbed 3–manifolds. It would be desirable to have a surgery exact sequence which does not mix them, and allows inductively the computation of each ℍ^{∗}(Γ, [k]) for each [k] independently.

The second motivation is the main result of [BN10]. This is a surgery formula for the Seiberg–Witten invariant of negative definite plumbed 3–manifolds; it compares these invariants for Γ and Γ \ j_{0} for fixed (non–mixed) spin^{c}–structures. The third term in the main formula of [BN10] comes from a ‘topological’ Poincaré series associated with the plumbing graph, and its nature is rather different than the other two terms.

Here our goal is to determine an exact sequence connecting ℍ^{∗}(Γ, [k̄]) and ℍ^{∗}(Γ\j_{0}, [R(k̄)]) (where R(k̄) is the restriction of k̄, see 4.1) with the newly defined third term, playing the role of the relative cohomology. Its relationship with the Poincaré series used in [BN10] will also be treated.

7.2. The ‘relative’ complex and cohomology. We consider a non–degenerate graph Γ and j_{0} one of its vertices.

We fix [k] ∈ Char(Γ \ j_{0})/2L(Γ \ j_{0}) and a characteristic element k_{m} ∈ [k] with w(k_{m}) = min_{k∈[k]} w(k). Furthermore, we fix a_{0} satisfying a_{0} ≡ e_{j_{0}} (mod 2). Then k_{a_{0}} := i(k_{m}) + ( (i(k_{m}), E_{j_{0}}) + a_{0} )E_{j_{0}}^{∗} ∈ Char(Γ). In fact, for any k′ ∈ [k] one gets that i(k′) + ( (i(k_{m}), E_{j_{0}}) + a_{0} )E_{j_{0}}^{∗} ∈ Char(Γ) and it is an element of [k_{a_{0}}]. For simplicity we write r_{0} for (i(k_{m}), E_{j_{0}}) + a_{0}.

We define

B_{0,rel} : F_{0}(Γ \ j_{0}, [k]) → F_{0}(Γ, [k_{a_{0}}])

by

(7.2.1)  B_{0,rel}(k′) = i(k′) + ( (i(k_{m}), E_{j_{0}}) + a_{0} )E_{j_{0}}^{∗} = i(k′) + r_{0}E_{j_{0}}^{∗}.

This extends to the level of complexes

B_{∗,rel} : (F_{∗}(Γ \ j_{0}, [k]), ∂) → (F_{∗}(Γ, [k_{a_{0}}]), ∂)

by B_{∗,rel}((k, I)) = (B_{0,rel}(k), I). Its dual

B_{rel}^{∗} : (F^{∗}(Γ, [k_{a_{0}}]), δ) → (F^{∗}(Γ \ j_{0}, [k]), δ)

is defined by B_{rel}^{∗}(ϕ) = ϕ ∘ B_{∗,rel}.

By a similar argument as in (5.3) we get that

B_{rel}^{∗} : F^{∗}(Γ, [k_{a_{0}}]) → F^{∗}(Γ \ j_{0}, [k]) is surjective.

We define the ‘relative’ complex F_{rel}^{∗} = F_{rel}^{∗}(Γ, j_{0}, [k], a_{0}]) via ker(B_{rel}^{∗}). Let the cohomology of the complex (F_{rel}^{∗}, δ) be ℍ_{rel}^{∗} = ℍ_{rel}^{∗}(Γ, j_{0}, [k], a_{0}). It is a graded ℤ[U]–module. We refer to it as the relative lattice cohomology.

Note that both F_{rel}^{∗} and B_{rel}^{∗} depend on the choice of the representative k_{a_{0}} of [k_{a_{0}}] and are not invariants merely of the classes [k_{a_{0}}] and [k].

Theorem 7.2.2. One has the short exact sequence of complexes

0 ⟶ F_{rel}^{∗} ⟶^{A_{rel}^{∗}} F^{∗}(Γ, [k_{a_{0}}]) ⟶^{B_{rel}^{∗}} F^{∗}(Γ \ j_{0}, [k]) ⟶ 0,

which provides a long exact sequence of ℤ[U]–modules:

⋯ ⟶ ℍ_{rel}^{q} ⟶^{𝔸_{rel}^{q}} ℍ^{q}(Γ, [k_{a_{0}}]) ⟶^{𝔹_{rel}^{q}} ℍ^{q}(Γ \ j_{0}, [k]) ⟶^{ℂ_{rel}^{q}} ℍ_{rel}^{q+1} ⟶ ⋯

(Again, the relative cohomology modules and the operators in the above exact sequence depend on the choice of k_{a_{0}}, and not only on its class [k_{a_{0}}].)

In the case when Γ is negative definite, the restriction of 𝔹_{rel}^{0} to T_{d[k_{a_{0}} ]}^{+} has its image in T_{d[k]}^{+}.

Proposition 7.2.3. Assume that Γ (but not necessarily Γ_{j_{0}}^{+}) is negative definite. Then

𝔹_{rel}^{0} : T_{d[k_{a_{0}} ]}^{+} → T_{d[k]}^{+} is onto.

In particular, ℍ_{rel}^{∗} has finite rank over ℤ. Moreover, one has an exact sequence of finite ℤ–modules:

0 ⟶ ℍ_{rel}^{0} ⟶ ℍ_{red}^{0}(Γ, [k_{a_{0}}]) ⊕ ℤ^{n} ⟶ ℍ_{red}^{0}(Γ \ j_{0}, [k]) ⟶
ℍ_{rel}^{1} ⟶ ℍ_{red}^{1}(Γ, [k_{a_{0}}]) ⟶ ℍ_{red}^{1}(Γ \ j_{0}, [k]) ⋯

where n := w_{Γ}(i(k_{m}) + r_{0}E_{j_{0}}^{∗}) − min{ w_{Γ} | [k_{a_{0}}]} ∈ ℤ_{≥0}.

Proof. First note that

w_{Γ}(i(k′) + r_{0}E_{j_{0}}^{∗}) = w_{Γ\j_{0}}(k′) − \frac{1 + r_{0}^{2}(E_{j_{0}}^{∗})^{2}}{8} .

This applied for k′ = k_{m} provides

(7.2.4)  n + \frac{d[k_{a_{0}}]}{2} = \frac{d[k]}{2} − \frac{1 + r_{0}^{2}(E_{j_{0}}^{∗})^{2}}{8} .

For any l ≥ 0 and k̄ ∈ [k_{a_{0}}] set ϕ̄_{l} ∈ F^{0}(Γ, [k_{a_{0}}]) defined by

ϕ̄_{l}(k̄) = U^{−l−d[k_{a_{0}} ]/2+w_{L}(k̄)} (k̄ ∈ [k_{a_{0}}]).

For different l ≥ 0 they generate T_{d[k_{a_{0}} ]}^{+} ⊂ ℍ^{0}(Γ, [k_{a_{0}}]). Similarly, set {ϕ_{l}}_{l≥0} in F^{0}(Γ \ j_{0}, [k]), where

ϕ_{l}(k′) = U^{−l−d[k]/2+w(k′)} (k′ ∈ [k]).

They generate T_{d[k]}^{+}. From the above identities and from the definition of B_{rel}^{0} one gets for any l ≥ 0:

𝔹_{rel}^{0}(ϕ̄_{n+l}) = ϕ_{l}.

Hence the restriction 𝔹_{rel}^{0} : T_{d[k_{a_{0}} ]}^{+} → T_{d[k]}^{+} is onto and the ℤ–rank of its kernel is n. □

The reader is invited to recall the definition of the Euler characteristic of the lattice cohomology from (2.3.5). We define the Euler characteristic of the relative lattice cohomology by

eu(ℍ_{rel}^{∗}) := ∑_{q≥0}(−1)^{q} rank_{ℤ} ℍ_{rel}^{q}.

Then the exact sequence of Proposition 7.2.3 and equation (7.2.4) provide

Corollary 7.2.5. With the notation r_{0} := (i(k_{m}), E_{j_{0}}) + a_{0} one has

eu(ℍ_{rel}^{∗}(Γ, j_{0}, [k], a_{0})) = eu(ℍ^{∗}(Γ, [k_{a_{0}}])) − eu(ℍ^{∗}(Γ \ j_{0}, [k])) − \frac{1 + r_{0}^{2}(E_{j_{0}}^{∗})^{2}}{8} .

Fix any l ∈ L(Γ \ j_{0}), then k_{a_{0}} + 2l = i(k_{m} + 2l) + r_{0}E_{j_{0}}^{∗}, hence we also get

Corollary 7.2.6. For any l ∈ L(Γ \ j_{0}) one has

\begin{matrix}eu(ℍ_{rel}^{∗}(Γ, j_{0}, [k_{m}], a_{0})) = eu(ℍ^{∗}(Γ, [k_{a_{0}}])) − \frac{(k_{a_{0}} + 2l)_{Γ}^{2} + |J |}{8} \\ −eu(ℍ^{∗}(Γ \ j_{0}, [k_{m}])) + \frac{(k_{m} + 2l)_{Γ\j_{0}}^{2} + |J \ j_{0}|}{8}. \end{matrix} ; (7.2.7)

7.3. Reinterpretation. Above, we started with an element k_{m} of the class [k], and we constructed one of its extensions k_{a_{0}}. Since k_{a_{0}} + i(2l) = i(k_{m} + 2l) + r_{0}E_{j_{0}}^{∗}, we have k_{m} + 2l = R(k_{a_{0}} + 2l) for any l ∈ L(Γ \ j_{0}).

This procedure can be inverted. Indeed, let us fix any k̄ ∈ Char(Γ) (which plays the role of k_{a_{0}}+i(2l)). Then set R(k̄) and define r_{0} by the identity r_{0}E_{j_{0}}^{∗} = k̄−iR(k̄). Finally, define

B_{0,rel} : F_{0}(Γ \ j_{0}, [R(k̄)]) → F_{0}(Γ, [k̄]) by B_{0,rel}(k′) = i(k′) + r_{0}E_{j_{0}}^{∗},

whose kernel is the relative complex F_{rel}^{∗} with cohomology ℍ_{rel}^{∗}(Γ, j_{0}, k̄). Then (7.2.7) reads as

\begin{aligned}eu(ℍ_{rel}^{∗}(Γ, j_{0}, k̄)) & = eu(ℍ^{∗}(Γ, [k̄])) − \frac{(k̄)_{Γ}^{2} + |J |}{8} \\ & − eu(ℍ^{∗}(Γ \ j_{0}, [R(k̄)])) + \frac{(R(k̄))_{Γ\j_{0}}^{2} + |J \ j_{0}|}{8}. \end{aligned} ; (7.3.1)

The identity (7.3.1) depends essentially on the choice of the choice of k̄ ∈ Char(Γ). In fact, even if we fix the class [k̄], the choice of the representative k̄ from the class [k̄] provides essentially different identities of type (7.3.1): not only the terms eu(ℍ_{rel}^{∗}(Γ, j_{0}, k̄)), (k̄)_{Γ}^{2} and (R(k̄))_{Γ\j_{0}}^{2} depend on the choice of k̄, but even the class [R(k̄)].

7.4. The connection with the topological Poincaré series. Let K_{Γ} ∈ L′ denote the canonical characteristic element of Γ defined by the adjunction formulae (K_{Γ} + E_{j}, E_{j}) + 2 = 0 for all j ∈ J. Similarly, one defines K_{Γ\j_{0}} ∈ Char(Γ \ j_{0}). Note that Char(Γ) = K + 2L′(Γ) and K_{Γ\j_{0}} = R(K_{Γ}). The next result computes eu(ℍ_{rel}^{∗}(Γ, j_{0}, K + 2l′)) in terms of l′ via the coefficients of a series associated with Γ.

Consider the multi-variable Taylor expansion Z(t) = ∑ p_{l′}t^{l′} at the origin of

(7.4.1)  ∏_{j∈J} (1 − t^{E_{j}^{∗}})^{δ_{j}−2},

where for any l′ = ∑_{j} l_{j}E_{j} ∈ L′ we write t^{l′} = ∏_{j} t_{j}^{l_{j}}, and δ_{j} is the valency of j. This lives in ℤ[[L′]], the submodule of formal power series ℤ[[t^{±1/d}]] in variables {t_{j}^{±1/d}}_{j}, where d = det(Γ). The series Z(t) was used in several articles studying invariants of surface singularities, see [CDG04, CDG08, CHR04, N08b, N08c, N10] for different aspects.

For any series S(t) ∈ ℤ[[L′]], S(t) = ∑_{l′} c_{l′}t^{l′}, we have the natural decomposition

S = ∑_{h∈L′/L} S_{h}, where S_{h} := ∑_{l′ : [l′]=h} c_{l′}t^{l′}.

In particular, for any fixed class [l′] ∈ L′/L, one can consider the component Z_{[l′]}(t) of Z(t). In fact, see e.g. [N08b, (3.1.20)],

(7.4.2)  Z_{[l′]}(t) = \frac{1}{d} ∑ ; ρ∈(L′/L)ρ([l′])^{−1} · ∏_{j∈J} (1 − ρ([E_{j}^{∗}])t^{E_{j}^{∗}})^{δ_{j}−2},

where (L′/L) is the Pontrjagin dual of L′/L.

Furthermore, once the vertex j_{0} of Γ is fixed, for any class [l′] ∈ L′/L we set

H_{[l′],j_{0}}(t) := Z_{[l′]}(t)| _{{t_{j_{0}} =t^{d} \atop t_{j} =1 for j≠j_{0}}} ∈ ℤ[[t]].

Let S(t) = ∑_{i≥0} c_{i}t^{i} be a formal power series. Suppose that for some positive integer p, the expression ∑_{i=0}^{pn−1} c_{i} is a polynomial P_{p}(n) in the variable n. Then the constant term of P_{p}(n) is independent of p. We call this constant term the periodic constant of S and denote it by pc(S) (cf. [NO09]).

Proposition 7.4.3. Fix the vertex j_{0} of Γ and write H_{[l′],j_{0}}(t) as ∑_{i≥0} c_{i}t^{i}.

(a) If l′ = ∑_{j} a_{j}E_{j}^{∗} = ∑_{j} l′_{j}E_{j} ∈ L′(Γ) with all a_{j} sufficiently large then

∑_{i<dl′_{j_{0}}} c_{i} = eu(ℍ_{rel}^{∗}(Γ, j_{0}, K + 2l′)).

(b) Take l̄′ = ∑_{j} l̄′_{j}E_{j} ∈ L′(Γ) with l̄′_{j_{0}} ∈ [0, 1). Then

pc(H_{[l̄′],j_{0}}) = eu(ℍ_{rel}^{∗}(Γ, j_{0}, K + 2l̄′)).

Proof. Use (7.3.1) from above and the identities (3.2.7) and (3.2.13) from [N10]. □

7.5. The connection with the Seiberg–Witten invariants. Let M(Γ) be the oriented plumbed 3–manifold associated with Γ, and −M(Γ) the same 3– manifold with opposite orientation. In is known that the spin^{c}–structures of M(Γ) (and of −M(Γ) too) can be identified with Char/2L, see e.g. [N05, N08]. Let sw(M(Γ), [k̄]) be the Seiberg–Witten invariant of M(Γ) associated with the spin^{c}– structure [k̄]. Then, by Theorem B of the Introduction of [N10] for a negative definite graph Γ one has

(7.5.1)  eu(ℍ^{∗}(Γ, [k̄])) = sw(−M(Γ), [k̄]).

Hence the above statements can be reinterpreted in terms of Seiberg–Witten invariants as well.

Example 7.5.2. Let Γ be the next graph, where all the vertices have decoration −2 except the j_{0} vertex which has −3.

[figure: j_{0}]

det(Γ) = det(Γ \ j_{0}) = 1, hence for both graphs we have only one class.

Moreover, Γ\j_{0} is rational (an E_{8}–graph), hence min w_{Γ\j_{0}} = 0, ℍ_{red}^{∗}(Γ\j_{0}) = 0 and eu(ℍ^{∗}(Γ \ j_{0})) = 0.

On the other hand, Γ is minimally elliptic, min w_{Γ} = 0, ℍ_{red}^{∗}(Γ) = ℍ_{red}^{0}(Γ) = ℤ_{(0)}, the rank one ℤ–module concentrated at degree zero. Hence eu(ℍ^{∗}(Γ)) = 1.

By the long exact sequence, we get ℍ_{rel}^{q}(Γ, j_{0}, k̄) = 0 for any k̄ and q > 0.

It is easy to see that K_{Γ} = −E_{j_{0}}^{∗}, and (E_{j_{0}}^{∗})^{2} = −1. Therefore, if k̄ = K + 2l′, and r_{0} := −(k̄, E_{j_{0}}^{∗}) and l′_{j_{0}} := −(l′, E_{j_{0}}^{∗}), then r_{0} = 2l′_{j_{0}} − 1. By (7.2.5) or (7.3.1)

rank ℍ_{rel}^{0}(Γ, j_{0}, k̄) = eu(ℍ^{∗}(Γ, j_{0}, k̄)) = \frac{r_{0}^{2} + 7}{8} = 1 + \frac{l′_{j_{0}}(l′_{j_{0}} − 1)}{2}.

By a computation one obtains

H_{[0],j_{0}}(t) = \frac{1 − t^{6}}{(1 − t^{3})(1 − t^{2})(1 − t)} = \frac{1 − t + t^{2}}{(1 − t)^{2}} = 1 + t + 2t^{2} + 3t^{3} + 4t^{4} + · · · .

Then

∑_{i<l′_{j_{0}}} c_{i} = 1 + 1 + 2 + 3 + ⋯ + (l′_{j_{0}} − 1) = 1 + \frac{l′_{j_{0}}(l′_{j_{0}} − 1)}{2},

hence (7.4.3)(a) follows. In order to realize part (b), we have to take l̄′ with l̄′_{j_{0}} = 0, hence in this case eu(ℍ^{∗}(Γ, j_{0}, k̄)) = 1. But one also has

pc \frac{1 − t + t^{2}}{(1 − t)^{2}} = 1.

Example 7.5.3. Assume that Γ is a star–shaped graph with central vertex j_{0} and we fix k̄ = K_{Γ}.

Since all the connected components of Γ \ j_{0} are strings (i.e. rational graphs), for them (cf. [N08])

eu(ℍ^{∗}(Γ \ j_{0}, [K_{Γ\j_{0}}])) − \frac{(K_{Γ\j_{0}})^{2} + |J \ j_{0}|}{8} = 0.

Moreover, ℍ^{q}(Γ, [K]) = ℍ_{rel}^{q}(Γ, j_{0}, [K]) = 0 for q > 0, and

rank ℍ_{rel}^{0}(Γ, j_{0}, [K]) = eu(ℍ^{∗}(Γ, [K]) − \frac{K^{2} + |J |}{8}

= rank ℍ^{0}(Γ, [K]) − min w_{Γ} − \frac{K^{2} + |J |}{8}

= rank ℍ^{0}(Γ, [K]) − min χ_{K}.

(7.5.4)

This equals the periodic constant of H_{[0],j_{0}}(t) by [N08, BN10]. Moreover, if (X, o) is a weighted homogeneous normal surface singularity with minimal good resolution graph Γ, then its geometric genus p_{g} equals the last term of (7.5.4), cf. e.g. [NN04, N05]. Hence p_{g}(X, o) = rank ℍ_{rel}^{0}(Γ, j_{0}, [K]).

References

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[CDG04] Campillo, A., Delgado, F. and Gusein-Zade, S. M., Poincaré series of a rational surface singularity, Invent. Math. 155(1) (2004), 41–53.

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Two exact sequences for lattice cohomology 21

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A. Rényi Institute of Mathematics, 1053 Budapest, Reáltanoda u. 13-15, Hungary.

    E-mail address: nemethi@renyi.hu